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College of Engineering Institute of Engineers in Sri Lanka (IESL) EE3001 - Advanced Measurements Class Test Answer only Five Questions Time Allowed: Three Hours Q(1) (i) A continuous-time signal x(t) is shown in the following Figure.Sketch the signals for (a) x(t)[u(t) – u(t-2)] (b) x(t)(t-3/2) (ii) Find, from first principle, the Fourier transform of the signal (iii) If the Laplace transform of f(t) shown in Figure (a) is F(s), shown that G(s), the Laplace transform of g(t)shown in Figure (b) is given by Figure (a) Figure (b) Hint: Remember that Q(2) (i) Using the definition of z-transform, show that 𝑎𝑘 𝑢(𝑘) ⇔ 𝑧 𝑧−𝑎 (ii) Using the z-transform derived in (i), or otherwise, find the z-transform X(z) of the (iii) Using the z-transform pairs given in the z-transform table, find the inverse z-transform of (iv) Using the graphical method, find y(t) x(t)*h(t) where x(t) and h(t) are shown in the following figure. Q(3) (i) Discuss following errors in a measurement system (a) Gross error (b) Systematic Errors (c) Random Errors (ii) During a test the current was measured as 30.4 A, flowing in a resistor of 0.105. Later on it is discovered that the ammeter reading was low by 1.2% and the marked resistance was high by 0.3%. Determine the true value of power as a percentage of the power that was originally calculated. (iii) The speed of a hydraulic turbine which is running at 1000 rpm is measured by a tachometer. The hydraulic turbine is subjected to variations in speed. For a sample of 20 readings at this speed, how many readings would you expect between 990 rpm and 1010 rpm. Assume that the tachometer gives a normal set of deviations with precision index of 0.04 at 1000rpm. Q(4) (i) Linear Time Invariant (LTI) system is specified by system equation (a) Find the characteristic polynomial, characteristic equation, characteristic roots and characteristic modes ofthis system. (b) Find y0(t), the zero-input component of the response y(t) for t ≥ 0, if the initial conditions arey0(0) = 1 and 𝑦0̇ (0) = 1 (ii) Given the initial conditions y0(0) 0 and 𝑦0̇ (0) = 1, find the unit impulse response of a LTI system specified by the equation Q(5) (a) State Nyquist Sampling Theorem (b) A discrete-time LTI system with a sampling frequency of 8kHz is shown in following Figure. Therectangular boxes with the label z -1 provide one sample period delay to their input signals.Thecircular components are adders or subtractors. The triangular components provide lineargain factors of ai or bi, where i is 0, 1 or 2. (i) Find the difference equation relating the output y[n] and input x[n]. (ii) Derive the system transfer function H(z). (iii) Given that the gain values are Find the poles and zeros of this system. Q(6) (i) Draw a circuit diagram to show the structure of a four bit 'R-2R ladder' type ofdigital to analogueconverter (DAC). Describe how a binary number is applied to the ladder network and explain how theappropriate analogue output voltage is generated (ii)4-bit R-2R DAC has VR=5 V, R= 5 kΩ. Calculate value of Vout for (a) LSB, 0001 (b) MSB, 1000 (iii) Determine the smallest step size (resolution) of a 4-bit ADC, which has a maximum output voltage of 12 V. Q(7) (i) The input-output relationship of a finite impulse response filter is given as (a) Write the z-transfer function of this filter (b) Show that the filter has a linear phase response (ii) (a) What do you understand by EMC ? (b) Describe followings related to Electro Magnetic Interference (EMI) (1) Inductive coupling (2) Capacitive Coupling (3) Mitigation of EMI Z-Transform Table Probability Table